Yahtzee Jr.

I played a kid's game this weekend and thought I was quite clever in figuring out a way to get a slight edge. I ended up beating Andrew and his two children twice (probably mostly through luck) but I wanted to see how slight that edge was.

The game is a very simplified version of Yahtzee in that you have 5 dice and 3 rolls to make your best 'hand' which you put on the scoring chart. All you're collecting is sets. There are no straights or full houses or anything like that. (In fact the dice all had pictures on them so the concept of a straight is right out.) There are two twists compared to normal Yahtzee. First, the dice each have the 5 main characters on them that you're collecting and 1 wild value. Second, one of the wilds is actually an anti-wild. It counts as nothing and can't be rerolled at all.

My 'trick' was to find the die with the anti-wild on it and keep it if I had a choice. To make things clearer, assume I'm using standard D6s and the 1s are wild. One of the dice has the 1 scratched off and replaced by something else to represent the anti-wild. I make my first roll and it comes out 1-2-2-3-3. Should I collect 2s or 3s? Normally in Yahtzee the answer to that question depends on what you've filled in on your scorecard but it often doesn't really matter. You have the same odds of getting 5-of-a-kind with either 2s or 3s. In this variant, however, I should find the die with the anti-wild on it. If it's on one of the dice current showing a 2 I should keep the 2s. If it's on a die currently showing a 3 I should keep it. I'm always keeping the wild as well. What sort of edge am I getting by choosing this way as opposed to always rerolling the anti-wild?

Well, I get to reroll each of the dice twice. The first die is 1/3 to give me what I want on the first roll and 2/3*1/3=2/9 to give me what I want on the second roll for a total of 5/9 to turn out good. The odds are the same for the second die. So, my overall odds are 16/81 - 3-of-a-kind. 40/81 - 4-of-a-kind. 25/81 - 5-of-a-kind.

What if I keep the anti-wild in play? The first die has the same odds as above but the second one is 1/6 to give me what I want on the first roll and 4/6*1/6=1/9 to give me what I want on the second roll. Overall that's 5/18 to give me what I want as opposed to 5/9 with the normal die. My overall odds here are 52/162 - 3-of-a-kind. 85/162 - 4-of-a-kind. 25/162 - 5-of-a-kind.

My expected value score in the first case is 4.1111. My expected value in the second case is 3.8333.

How about if my first roll is worse? (Simplify things a little and assume you can't switch horses in mid stream, so even if you roll 3-3-3-3 on your second roll it doesn't help if you kept a 2 to start.) 2-3-4-5-6, say? Pick out the bad die and keep it and you end up with 256/6561 - 1-of-a-kind. 1280/6561 - 2-of-a-kind. 2400/6561 - 3-of-a-kind. 2000/6561 - 4-of-a-kind. 625/6561 - 5-of-a-kind. Run with something else and you end up with 832/13122 - 1-of-a-kind. 3440/13122 - 2-of-a-kind. 5100/13122 - 3-of-a-kind. 3125/13122 - 4-of-a-kind. 625/13122 - 5-of-a-kind.

My expected value in the first case is 3.2222. My expected value in the second case is 2.9444. I am happy but not too surprised to see the difference is the same. A gain a little more than a quarter of a point by making this choice over making the wrong choice. Someone who isn't aware of this is going to pick right half the time, so my edge is .1389 points. This is, of course, assuming that I didn't roll the anti-wild in my first roll and that the die it's on is one of my potential best picks.

Or does it? What if I roll 2-2-3-4-5? Should I keep the 3 if it's the anti-wild? I already know my expected value in that case, so I need to work out the results when I keep the 2-2. Well, my odds are 208/1458 - 2-of-a-kind. 600/1458 - 3-of-a-kind. 525/1458 - 4-of-a-kind. 125/1458 - 5-of-a-kind. Or an expected value of 3.3889. This is bigger than the 3.2222 where I keep just the anti-wild die, so no unexpected diversions in strategy here. Keep the bigger raw number, only use the location of the anti-wild to break ties. This was my gut feeling at the time, so I'm glad it pans out in reality.

Now the real question is should I really be looking for edges in order to beat 6 and 7 year old kids at board games? Now that I have the edge, should I use it? Should I try to explain it to them? Should I only use it if Andrew is also playing?